Equilibrium constants reduction half-reactions

One of the most useful applications of standard potentials is in the calculation of equilibrium constants from electrochemical data. The techniques that we develop here can be applied to any kind of reaction, including neutralization and precipitation reactions as well as redox reactions, provided that they can be expressed as the difference of two reduction half-reactions. 

Table 7.1 Standard Electrode Potentials and Equilibrium Constants for Some Reduction Half-Reactions. ...

Table 4.2 Equilibrium constants of reduction half-reactions at pH 7 and 25 °C...

Table 7.7 Equilibrium constants of reduction half-reactions of trace elements in submerged soils compared with those for Fe and Mn...

The equilibrium constant for this reaction is actually the solubility product, Ksp, for silver chloride (Section 11.10). It does not matter that overall the reaction is not a redox reaction so long as it can be expressed as the differ- ence of two reduction half-reactions. Because silver chloride is almost insol-i uble, we expect K to be very small (and E° to be negative). 

Standard half-cell potentials can be used to compute free energy changes and equilibrium constants for many reactions other than oxidation-reduction reactions. 

The effects of pH on the standard apparent reduction potentials of the half reactions involved in the nitrogenase reaction are shown in Table 9.5. The effects of pH on the apparent equilibrium constants of the reactions involved in the nitrogenase reaction as shown in Table 9.6. 

In soil solutions the most important chemical elements that undergo redox reactions are C, N, O, S, Mn, and Fe. For contaminated soils the elements As, Se, Cr, Hg, and Pb could be added. Table 2.4 lists reduction half-reactions (most of which are heterogeneous) and their equilibrium constants at 298.15 K under 1 atm pressure for the six principal elements involved in soil redox phenomena. Although the reactions listed in the table are not full redox reactions, their equilibrium constants have thermodynamic significance and may he calculated with the help of Standard-State chemical potentials in the manner... 

Note that, when the iron half-reaction is doubled, its electrode potential is unaffected because the potential does not depend on the number of ions involved. Thus a Nemst equation written for a doubled reaction is identical to that for a single reaction. If the subtraction is carried out in such a direction as to produce a positive cell emf, the equilibrium constant for the reaction is greater than unity, and the reaction is said to be spontaneous. This conclusion follows because the stronger oxidant [in this case Fe(III)] has the higher potential, and hence the subtraction is performed in the direction corresponding to the reduction of the stronger oxidant. 

Before we discuss redox titration curves based on reduction-oxidation potentials, we need to learn how to calculate equilibrium constants for redox reactions from the half-reaction potentials. The reaction equilibrium constant is used in calculating equilibrium concentrations at the equivalence point, in order to calculate the equivalence point potential. Recall from Chapter 12 that since a cell voltage is zero at reaction equilibrium, the difference between the two half-reaction potentials is zero (or the two potentials are equal), and the Nemst equations for the halfreactions can be equated. When the equations are combined, the log term is that of the equilibrium constant expression for the reaction (see Equation 12.20), and a numerical value can be calculated for the equilibrium constant. This is a consequence of the relationship between the free energy and the equilibrium constant of a reaction. Recall from Equation 6.10 that AG° = —RT In K. Since AG° = —nFE° for the reaction, then... 

In a spontaneous reaction between two half-ceUs, the half-ceU with the more positive potential in Table 2.2 undergoes reduction, while the one with the more negative potential undergoes oxidation. The charge transfer changes the standard electrode potential due to the change of the composition of the electroactive species in the electrolyte. When a final ratio of the activities of the reactive species, as defined in Eq. (2.38), is equal to the equilibrium constant of the reaction, the system will be in equilibrium. 

A chemical reaction does not have to be an oxidation-reduction reaction for us to calculate its equilibrium constant. It is only necessary to be able to write the reaction as the sum of an oxidation half-reaction and a reduction half-reaction. 

C60 with strong acids, a C q solution was generated by controlled potential electrolysis and titrated with concentrated triflic acid. UV/visible-near IR spectra as well as steady state voltammetry with ultramicroelectrodes were used to monitor 50 and C60H. Addition of the triflic acid led to a decrease of absorbance and steady state oxidation current of Cgo". At the same time, a reduction current appeared at potentials more negative than the half wave potential of Cgg oxidation. This current was attributed to CeoH reduction. With some simple assumptions, the current changes were used to calculate the equilibrium constant of the reaction ... 

In addition to defined standard conditions and a reference potential, tabulated half-reactions have a defined reference direction. As the double arrow in the previous equation indicates, E ° values for half-reactions refer to electrode equilibria. Just as the value of an equilibrium constant depends on the direction in which the equilibrium reaction is written, the values of S ° depend on whether electrons are reactants or products. For half-reactions, the conventional reference direction is reduction, with electrons always appearing as reactants. Thus, each tabulated E ° value for a half-reaction is a standard reduction potential. 

This is a quantitative calculation, so it is appropriate to use the seven-step problem-solving strategy. We are asked to determine an equilibrium constant from standard reduction potentials. Visualizing the problem involves breaking the redox reaction into its two half-reactions ... 

The voltage for a complete reaction is the difference between the potentials of the two half-reactions E = E+ — E, where E+ is the potential of the half-cell connected to the positive terminal of the potentiometer and E is the potential of the half-cell connected to the negative terminal. The potential of each half-reaction is given by the Nemst equation E = E° — (0.059 16/n) log Q (at 25°C), where each reaction is written as a reduction and Q is the reaction quotient. The reaction quotient has the same form as the equilibrium constant, but it is evaluated with concentrations existing at the time of interest. Electrons flow through the circuit from the electrode with the more negative potential to the electrode with the more positive potential. 

Because of the bulk of comparable material available, it has been possible to use half-wave potentials for some types of linear free energy relationships that have not been used in connection with rate and equilibrium constants. For example, it has been shown (7, 777) that the effects of substituents on quinone rings on their reactivity towards oxidation-reduction reactions, can be approximately expressed by Hammett substituent constants a. The susceptibility of the reactivity of a cyclic system to substitution in various positions can be expressed quantitatively (7). The numbers on formulae XIII—XV give the reaction constants Qn, r for the given position (values in brackets only very approximate) ... 

When a biochemical half-reaction involves the production or consumption of hydrogen ions, the electrode potential depends on the pH. When reactants are weak acids or bases, the pH dependence may be complicated, but this dependence can be calculated if the pKs of both the oxidized and reduced reactants are known. Standard apparent reduction potentials E ° have been determined for a number of oxidation-reduction reactions of biochemical interest at various pH values, but the E ° values for many more biochemical reactions can be calculated from ArG ° values of reactants from the measured apparent equilibrium constants K. Some biochemical redox reactions can be studied potentiometrically, but often reversibility cannot be obtained. Therefore a great deal of the information on reduction potentials in this chapter has come from measurements of apparent equilibrium constants. 

Since tables of standard apparent reduction potentials and standard transformed Gibbs energies of formation contain the same basic information, there is a question as to whether this chapter is really needed. However, the consideration of standard apparent reduction potentials provides a more global view of the driving forces in redox reactions. There are two contributions to the apparent equilibrium constant for a biochemical redox reaction, namely the standard apparent reduction potentials of the two half-reactions. Therefore it is of interest to compare the standard apparent reduction potentials of various half reactions. 

The Tafel expressions for both the anodic and the cathodic reaction can be directly incorporated into a mixed potential model. In modeling terms, a Tafel relationship can be defined in terms of the Tafel slope (b), the equilibrium potential for the specific half-reaction ( e), and the exchange current density (70), where the latter can be easily expressed as a rate constant, k. An attempt to illustrate this is shown in Fig. 10 using the corrosion of Cu in neutral aerated chloride solutions as an example. The equilibrium potential is calculated from the Nernst equation e.g., for the 02 reduction reaction,... 

Although the law of mass action is equally valid for oxidation-reduction processes, and therefore conclusions as to the direction of reactions may be drawn from the knowledge of equilibrium constants, traditionally a different approach is used for such processes. This has both historical and practical reasons. As pointed out in the previous sections, in oxidation-reduction processes electrons are transferred from one species to another. This transfer may occur directly, i.e. one ion collides with another and during this the electron is passed on from one ion to the other. It is possible, however, to pass these electrons through electrodes and leads from one ion to the other. A suitable device in which this can be achieved is a galvanic cell, one of which is shown in Fig. 1.14. A galvanic cell consists of two half-cells, each made up of an electrode and an electrolyte. The two electrolytes are connected with a salt bridge and, if... 

The third largest class of enzymes is the oxidoreductases, which transfer electrons. Oxidoreductase reactions are different from other reactions in that they can be divided into two or more half reactions. Usually there are only two half reactions, but the methane monooxygenase reaction can be divided into three "half reactions." Each chemical half reaction makes an independent contribution to the equilibrium constant E for a chemical redox reaction. For chemical reactions the standard reduction potentials ° can be determined for half reactions by using electrochemical cells, and these measurements have provided most of the information on standard chemical thermodynamic properties of ions. This research has been restricted to rather simple reactions for which electrode reactions are reversible on platinized platinum or other metal electrodes. 

When the pH is specified, each biochemical half reaction makes an independent contribution to the apparent equilibrium constant K for the reaction written in terms of reactants rather than species. The studies of electochemical cells have played an important role in the development of biochemical thermodynamics, as indicated by the outstanding studies by W. Mansfield Clarke (1). The main source of tables of ° values for biochemical half reactions has been those of Segel (2). Although standard apparent reduction potentials ° can be measured for some half reactions of biochemical interest, their direct determination is usually not feasible because of the lack of reversibility of the electrode reactions. However, standard apparent reduction potentials can be calculated from for oxidoreductase reactions. Goldberg and coworkers (3) have compiled and evaluated the experimental determinations of apparent equilibrium constants and standard transformed enthalpies of oxidoreductase reactions, and their tables have made it possible to calculate ° values for about 60 half reactions as functions of pH and ionic strength at 298.15 K (4-8). 

Reactions that take place consecutive to the electrode process can be studied polarographioally only in those cases in which the electrode process is reversible. In these cases the wave-heights and the wave-shape remain unaffected by the chemical processes. However, the half-wave potentials are shifted relative to the equilibrium oxidation-reduction potential, determined e.g. potentiometrically. Hence, whereas in all above examples, limiting currents were measured to determine the rate constant, it is the shifts of half-wave potentials which are measured here. First- and second-order chemical reactions will be discussed in the following. 

The foregoing example illustrates how equilibrium constants for overall cell reactions can be determined electrochemically. Although the example dealt with redox equilibrium, related procedures can be used to measure the solubility product constants of sparingly soluble ionic compounds or the ionization constants of weak acids and bases. Suppose that the solubility product constant of AgCl is to be determined by means of an electrochemical cell. One half-cell contains solid AgCl and Ag metal in equilibrium with a known concentration of CP (aq) (established with 0.00100 M NaCl, for example) so that an unknown but definite concentration of Kg aq) is present. A silver electrode is used so that the half-cell reaction involved is either the reduction of Ag (aq) or the oxidation of Ag. This is, in effect, an Ag" Ag half-cell whose potential is to be determined. The second half-cell can be any whose potential is accurately known, and its choice is a matter of convenience. In the following example, the second half-cell is a standard H30" H2 half-cell. 

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